Volume 24, 2020
|Page(s)||408 - 434|
|Published online||25 September 2020|
Empirical measures: regularity is a counter-curse to dimensionality
Université Paris-Est, Laboratoire d’Analyse et de Matématiques Appliquées (UMR 8050), UPEM, UPEC, CNRS,
* Corresponding author: email@example.com
Accepted: 6 November 2019
We propose a “decomposition method” to prove non-asymptotic bound for the convergence of empirical measures in various dual norms. The main point is to show that if one measures convergence in duality with sufficiently regular observables, the convergence is much faster than for, say, merely Lipschitz observables. Actually, assuming s derivatives with s > d∕2 (d the dimension) ensures an optimal rate of convergence of 1/√n (n the number of samples). The method is flexible enough to apply to Markov chains which satisfy a geometric contraction hypothesis, assuming neither stationarity nor reversibility, with the same convergence speed up to a power of logarithm factor. Our results are stated as controls of the expected distance between the empirical measure and its limit, but we explain briefly how the classical method of bounded difference can be used to deduce concentration estimates.
Mathematics Subject Classification: 60B10 / 60J05 / 62E17 / 49Q20
Key words: Concentration / dual norms / empirical measure / Markov chains / non-asymptotic bounds / Wasserstein metric
© The authors. Published by EDP Sciences, SMAI 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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